Wednesday, December 18, 2013

This closed curve looks a bit like a lateral slice of a cerebral cortex, and has a very simple parametric formula:

If you let n = 0, the curve is a circle, for n = 1, you get an epicycloid, and for n around 4 you start to get something that looks like the brain slice. I think that there is a small copy of the whole curve on the left, giving the curve some self-similarity.

I first saw this curve, and some other neat ones that have very similar formulas, when looking at the scrambler amusement park ride (see here, and here).

Update: A very nice Mathematica animation that traces out this curve can be found at Curiosa Mathematica.

Tuesday, November 26, 2013

A nice post on mathen inspired me to construct some Euler spirals using Fathom and Tinkerplots. I like using these tools for this sort of playing around - they are intended for middle and high-school data management activities, but are, effectively, simple quasi-programming environments. The results are not as pretty as those from mathen, but are nice enough and very easy to generate.

To recreate images like these in either Fathom or Tinkerplots you need two sliders - I called them magnitude and delta.

You then create a collection (or card set in TP) with four attributes (n, x, y, and theta). Each attribute will have its data generated by a formula, as shown below.

Adding cases to the collection generates the data (the pictures shown have about 2000 cases). You can then graph or plot the results using the x and y attributes as your axes. Varying magnitude adjusts the gap between the points, varying delta adjusts how quickly the curvature changes.

Some values for delta suggested that you can obtain curves within curves and fractal-like images.

Tuesday, November 19, 2013

Here are instructions for making a snowflake iteration similar to the one shown in the previous post using Geometer's Sketchpad. I generally don't put useful things like instructions on this blog, but I thought I would make an exception: these fun and easy constructions are worth doing because they are pretty, and they illustrate some important concepts associated with recursion. GSP is a great tool for playing with these sorts of things - I haven't explored Geogebra at all, so I can't comment on what similar sorts of things can be done with that tool.

1. Draw a line segment AB (using the segment tool), construct its midpoint C (with the line selected, choose Construct > Midpoint), and then construct the midpoint of AC (we'll call that D).

The important thing to note is that AB is the only thing that you will draw. Everything else will be constructed. What we are doing here is a really good example of what GSP aficionados call geometric programming: AB is your input, and everything else we construct as part of a program written in the language of geometry. A mistake that some people make with GSP when they are first playing with it is to consider it a drawing rather than construction tool - you really should draw very little, and construct a lot.

2. Mark A as the center of rotation (select A, and then Transform > Mark Center) and rotate all lines and points around A by 60 degrees, five times. You'll end up with six spokes that look like this:

3. Connect the midpoints of the spokes to form a hexagon.

4. Select all sides of the hexagon and construct the midpoints of the sides. Then connect those midpoints to the points on the interior of the hexagon to form a star, like the one shown below.

5. At this point it would be wise to hide some of the things that we don't want to include in our construction (select lines and dots and press CNTRL+H). You should leave your star, the original points A and B, and the mid and end-points of each spoke.

6. Now we are ready to iterate. To do this, select the points A and B, and then open the Iterate dialog (Transform > Iterate...). The first iteration is to map A to C and B to itself. Keep the dialog open (we need to add more maps to this iteration).

7. Repeat the same sort of mapping on each spoke: A gets mapped to the midpoint of the spoke, and B gets mapped to the endpoint of the spoke. Do this by using the Struct... > Add New Map on the iteration dialog.

8. After mapping onto all the spokes, hit the Iterate button.

9. You can now hide any points or parts of the iteration that you'd like, and increase or decrease the number of iterations (select the whole shape and use the plus (+) and minus (-) keys).

Two thing to note: when we mapped A and B, we mapped them onto points that were derived from A and B (midpoints and endpoints of the line AB, and rotations of those points), and we mapped A and B such that the distance between their images was smaller than the original distance between A and B. Mapping A and B so that they moved closer to each other means that the resulting shape will be bounded - otherwise the shape will get larger with each iteration.

Monday, November 11, 2013

Going through some old files, I found this bunch of GSP iterations based on a square - very similar to this post from 2 years ago today. These reminded me of the designs in Wucius Wong's 1972 book Principles of Two-Dimensional Design - previously mentioned here.

Saturday, November 9, 2013

A favorite counting puzzle is finding the total number of paths through some diagram. In puzzles like the one above, you are asked to find how many different paths will take you from one end of the diagram to the other, always following the direction of the arrows. To count the paths easily, you should apply the rule of sum and the rule of product.

In the diagram above, the rule of sum comes into play in getting around a single square. You can go around the top (1 choice) or around the bottom (1 choice), this gives you 1+1 = 2 choices. Getting around the first square and the second square and the next square, etc. requires the rule of product: 2 x 2 x 2 x 2 x 2 = 32 paths. In path puzzles like this, you add "or" choices and multiply "and" choices.

Wednesday, October 23, 2013

In the introduction to Wheels, Life and Other Mathematical Amusements (one of the 15 collections of his Mathematical Games articles for Scientific American), Martin Gardner sums up most of what can be said about effective mathematics teaching and learning in a couple of lines:

Like science, mathematics is a kind of game that we play with the universe. The best mathematicians and the best teachers of mathematics obviously are those who both understand the rules of the game, and who relish the excitement of playing it.

If you are reading this blog, you've probably heard of Gardner before: anyone who enjoys recreational mathematics is bound to stumble upon Gardner's books (a near-inexhaustible resource), and his stature seems to have only grown in recent years despite the fact that he stopped writing his influential Mathematical Games column in the early 1980's.

Something new for Gardner's many readers is his new autobiography Undiluted Hocus-Pocus, written just prior to his death in 2010. Gardner at times criticizes his own text as slovenly and disheveled, but its casual tone and associative leaps makes it an enjoyable read that conveys a strong sense of Gardner's personality.

Why is an autobiography of Martin Gardner of interest? I have to admit that what first pulled me in was curiosity about the many personalities that Gardner knew and could tell stories about. He was a friend of many eminent mathematicians and skeptics, and his work as a popularizer of math and science brought him into contact with other, more mainstream, media celebrities. Hocus Pocus (particularly chapters 14 through 17) covers this period nicely. The path of Gardner's surprisingly successful journalistic and writing career is also a worthy tale - his leap from writing limericks and designing puzzles for a children's magazine to writing in Scientific American [confirms how truth can be stranger than fiction].

Gardner's success is even more surprising than a quick look at his career would suggest. His interests were as far away from the cultural mainstream as you'd care to wander: magic, mathematics and skepticism. Even within those communities, Gardner seemed to be on the periphery: a non-performing magician; a self-educated mathematician who wrote popular accounts on recreational mathematics; a skeptic who believed in god. Even his literary interests tended towards the neglected and and unfashionable: Frank Baum, C.K. Chesterton, and Lewis Carroll were his literary icons. But despite lacking insider credentials and having unpopular interests, Gardner became an icon. In mathematics, his impact has been particularly significant: many working mathematicians today credit his writings as sparking and sustaining their interest in mathematics, and of the small fragments of mathematics that have made their way into popular culture, many were introduced to us by Gardner (Conway's Game of Life, Penrose tiles, Surreal numbers, the art of M.C. Escher, fractals, polyominos, to name a few).

Through Undiluted Hocus-Pocus, we get some clues that help explain the Gardner enigma. Gardner never claimed to be a mathematician, magician, or professional skeptic - only a journalist, and being a journalist and a writer is what he worked hard at. He broke that so-called first rule of writing: instead of writing what he knew, he wrote of what he wanted to know, which lead him not only to much of his mathematical writing, but also to write books on Relativity and the philosophy of science. He sought out and maintained many intellectual friendships and correspondences that brought him new ideas and topics. In following where his interests and friends led him, he seems to have encountered each new topic as a game - and the playfulness at the heart of much of his writing is the key element that hooked his readers and drew them in.

I was surprised to learn how strong and sustained interest in religion and theology remained throughout his life. The books he spends the most time describing are his religious-themed The Flight of Peter Fromm and his personal philosophical statement The Whys of a Philosophical Scriviner. Gardner's rational and skeptical views and his love of science made him a critic of organized religion, but his early and deep induction into religious belief seem to have held on in the form of a personal pragmatic theology, greatly influenced by the writings of William James (whom he references often). A last bastion of irrationality in Gardner's world view, and his insistence of the ultimate mysteriousness of consciousness, may put off readers who encountered him through his writings on science and skepticism; and the casual certainty that makes his expository mathematics writing so readable doesn't suit his philosophical commentary as well. Gardner's philosophical views are not incongruous with his love of mathematics, however - they are a reminder that mathematics provides people, whatever they believe about the world, a game that everyone can play. Undiluted Hocus-Pocus reminds us howGardner taught many of us how to play the game of mathematics better.

Thursday, September 26, 2013

An early post back in 2008 had to do with playing with knot tiles. Finding the old program for this, I started nesting the basic plait pattern in itself to create a border-effect. I liked how they turned out, so I decided to share a few.

Friday, September 20, 2013

I've just finished up the Stanford Online course How to Learn Math by Prof. Jo Boaler. A very interesting course, and a nice counterpoint to the heated railing against inquiry-based learning that is all the rage here in Ontario (see some of that here, and here).

One of the activities highlighted in the course was something Prof. Boaler called a "number talk," which can take many forms - one being a group discussion about strategies for multiplying. I've mentioned before that my elementary teachers mainly used threats to get us kids to memorize the multiplication tables - it's of course much better to memorize less, and instead rely on number-sense to reason out the answer. Do enough of this, and you do end up memorizing much of your times tables, but rather than being the goal it's an outcome of a much more worthwhile exploration of numbers and their properties.

These number talks reminded me of an article that appeared a while back on The Guardian Data Blog, which showed some data regarding the multiplication facts children have the most trouble with. It included a nice interactive graphic derived from having about 200 children answer some multiplication questions online. You can re-create the nice pictures from the article in Tinkerplots by importing the data from the blog post.

The plot below shows the 1-12x tables, with each cell shaded according to how many times students got the corresponding multiplication problem wrong. You can see the nice lightly coloured row and columns corresponding to the 10x table, and you can see where kids got into trouble in the middle of the tables.

I think it is interesting for teachers, and for students, to talk about what this plot shows about multiplication facts that kids seem to know, the properties they might be exploiting (the near symmetry shows that commutativity is not a major problem for these kids), and the strategies that might help them figure out the tougher-to-remember products. You can see that some facts are likely memorized (12 x 12 shouldn't really be easier than 12 x 11, but it seems to be better known). Although you can't tell here what strategies are being used or neglected, you can take a look at where things seem to get tougher and think about various strategies that could be used to get to the answers.

One question: what is the hardest product for students? If the shading above isn't clear, a scatter plot might help:

Surprisingly, it's 48 - apparently 6 x 8 and 8 x 6 are big confounders (I'm not sure which was tougher, the data didn't say explicitly which operand came first). Now how would you figure out 6 times 8 if you don't know it by heart? How about (5 + 1) 8 = 40 +8, or perhaps 2 x 3 x 8 = 2 x 24? See how useful the distributive property and factoring can be? Watching students reason out various strategies in the video presented in the "How to Learn Math" course showed how helpful discussing and dissecting these types of questions can be for kids.

Interestingly, Prof Boaler is launching a website and a non-profit organization (youcubed.org) to promote the teaching strategies and approaches to math education that she enthusiastically promoted in the course. It will be interesting to see where this math education entity fits in with all the other emerging voices that are calling for different ways of teaching mathematics (compare youcubed.org with JUMP and with Computer Based Math, for example).

Friday, June 28, 2013

The recursive formula used to generate the Mandelbrot set is quadratic - here are some variations that use different powers in a formula that is otherwise the same as the Mandelbrot formula.

A strange variation on the Mandelbrot theme is the burning ship fractal. It is generated in a similar way, but the first step in calculating each term is to take the "absolute value" of the previous term. Actually abs(z) is a complex number whose coefficients are the absolute values of the corresponding coefficients of z. Although not apparent in the picture below (see hpdz for deeper images) , just as in the fractals above there are small copies of the larger image (off to the left there, in the corner)

These variations on the burning ship use different powers with surprising results. Personally, I find these a bit sinister looking - each roughly polygonal form has strange organic looking out-growths and hidden small scale replicas of itself.

The last Mandelbrot variant shown here is sometimes known as the Tricorn or Mandelbar - so named because the first step in computing a term in the z_n sequence is to take the complex conjugate of the previous term (bar-z). Surprisingly you get three smeared copies of the Mandelbrot set radiating out of the vertices of what looks like a hypocycloid.